NCERT Class XI Mathematics - Sets - Solutions

Question : 72

Total: 73

In a survey of 60 people, it was found that 25 people read newspaper H, 26 read newspaper T, 26 read newspaper I, 9 read both H and I, 11 read both H and T, 8 read both T and I, 3 read all three newspapers. Find:
(i) the number of people who read at least one of the newspapers.
(ii) the number of people who read exactly one newspaper.

Solution:

n(U) = 60 ... (i)
n(H) = a + b + c + d = 25 ... (ii)
n(T) = b + c + f + g = 26 ... (iii)
n(I) = c + d + e + f = 26 ... (iv)
n(H ∩ I) = c + d = 9 ... (v)
n(H ∩ T) = b + c = 11 ... (vi)
n(T ∩ I) = c + f = 8 ... (vii)
n(H ∩ T ∩ I) = c = 3 ... (viii)

Putting value of c in (vii), 3 + f = 8 ⇒ f = 5
Putting value of c in (vi), 3 + b = 11 ⇒ b = 8
Putting value of c in (v), 3 + d = 9 ⇒ d = 6
Putting value of c, d, f in (iv), 3 + 6 + e + 5 = 26 ⇒ e = 26 – 14 = 12
Putting value of b, c, f in (iii), 8 + 3 + 5 + g = 26 ⇒ g = 26 – 16 = 10
Putting value of b, c, d in (ii), a + 8 + 3 + 6 = 25 ⇒ a = 25 – 17 = 8
(i) Number of people who read at least one of the three newspapers
= a + b + c + d + e + f + g
= 8 + 8 + 3 + 6 + 12 + 5 + 10 = 52
(ii) Number of people who read exactly one newspaper = a + e + g = 8 + 12 + 10 = 30.

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